There is a need to estimate the time it will take a missile to intercept a target or to arrive at the point of closest approach. The time of flight to intercept or to the point of closest approach is known as the time-to-go τ. The time-to-go is very important if the missile carries a warhead that should detonate when the missile is close to the target. Accurate detonation time is critical for a successful kill. Proportional navigation guidance does not explicitly require time-to-go, but the performance of the advanced guidance law depends explicitly on the time-to-go. The time-to-go can also be used to estimate the zero effort miss distance.
One method to estimate the flight time is to use a three degree of freedom missile flight simulation, but this is very time consuming. Another method is to iteratively estimate the time-to-go by assuming piece-wise constant positive acceleration for thrusting and piece-wise constant negative acceleration for coasting. Yet another method is to iteratively estimate the time-to-go based upon minimum-time trajectories.
Tom L. Riggs, Jr. proposed an optimal guidance method in his seminal paper “Linear Optimal Guidance for Short Range Air-to-Air Missiles” by (Proceedings of NAECON, Vol. II, Oakland, Mich., May 1979, pp. 757-764). Riggs' method used position, velocity, and a piece-wise constant acceleration to estimate the anticipated locations of a vehicle and a target/obstacle and then generated a guidance command for the vehicle based upon these anticipated locations. To ensure the guidance command was correct, Riggs' method repeatedly determined the positions, velocities, and piece-wise constant accelerations of both the vehicle and the target/obstacle and revised the guidance command as needed. Because Riggs' method did not consider actual, or real time acceleration in calculating the guidance command, a rapidly accelerating target/obstacle required Riggs' method to dramatically change the guidance command. As the magnitude of the guidance command is limited, (for example, a fin of a missile can only be turned so far) Riggs' method may miss a target that it was intended to hit, or hit an obstacle that it was intended to miss. Additionally, many vehicles and targets/obstacles can change direction due to changes in acceleration. Riggs' method, which provided for only piece-wise constant acceleration, may miss a target or hit an obstacle with constantly changing acceleration.
Computationally, the fastest methods use only missile-to-target range and range rate or velocity information. This method provides a reasonable estimate if the missile and target have constant velocities. When the missile and/or target have changing velocities, this simple method provides time-to-go estimates that are too inaccurate for warheads intended to detonate when the missile is close to the target.
FIG. 1 illustrates two different prior art methods for determining time-to-go. FIG. 1 shows a missile 100 with a net velocity v relative to the target at a missile-to-target angle relative to the LOS between the missile 100 and a target 104. The net velocity v is a function of both the missile 100 and the target 104 velocities. The missile-to-target range is shown as r. As such a target intercept scheme occurs in three-dimensional space, vectors will be shown in bold, while the magnitudes of such vectors will be shown as standard text.
Assuming the missile and target velocities are constant, the distance between the missile 100 and target 104 at time t is:z=r+vt.  Eq. 1The miss distance is minimized when
                                          ∂                          (                              z                ·                z                            )                                            ∂            t                          =        0.                            Eq        .                                  ⁢        2            Substituting Eq. 1 into Eq. 2 yields:r·v+v·vt=0.  Eq. 3Solving Eq. 3, the time-to-go τ is:
                    τ        =                  -                                                    v                ·                r                                            v                ·                v                                      .                                              Eq        .                                  ⁢        4            Eq. 4 yields the exact time-to-go if the missile 100 and target 104 have constant velocities.
The minimum missile-to-target position vector z can be obtained by substituting Eq. 4 into Eq. 1 resulting in:
                    z        =                                                                              (                                      v                    ·                    v                                    )                                ⁢                r                            -                                                (                                      v                    ·                    r                                    )                                ⁢                v                                                    v              ·              v                                =                                                                      (                                      v                    ×                    r                                    )                                ×                v                                            v                ·                v                                      .                                              Eq        .                                  ⁢        5            The zero-effort-miss distance, corresponding to the magnitude of the minimum missile-to-target position vector z, illustrated as point P in FIG. 1, is:
                                        z                          =                                                                                          (                                      v                    ×                    r                                    )                                ×                v                                            v                ·                v                                                          =                                                                      v                  2                                ⁢                r                ⁢                                                                  ⁢                sin                ⁢                                                                  ⁢                α                                            v                2                                      =                          r              ⁢                                                          ⁢              sin              ⁢                                                          ⁢                              α                .                                                                        Eq        .                                  ⁢        6            
The prior art time-to-go formulation is simply:
                              τ          =                      r                          r              .                                      ,                            Eq        .                                  ⁢        7            where {dot over (r)} is the range rate. The difference between Eq. 4 and Eq. 7 is apparent in FIG. 1. Eq. 4 estimates the flight time for the missile 100 to reach the point of closest approach, P. Eq. 7, however, estimates the flight time for the missile 100 to reach point Q. If the missile 100 and target 104 have no acceleration, then Eq. 4 is exact. However, if a missile guidance system is trying to align the relative velocity with the LOS, the missile 100 is likely to travel the range r. In this case, Eq. 7 is more appropriate for estimating the time-to-go. On the other hand, if zero-effort-miss distance is needed by the missile guidance system, Eq. 4 is more appropriate. It must be emphasized that Eqs. 4 and 7 are only accurate when both the target 104 and the missile 100 have constant velocities.
A simple technique that includes the effect of acceleration by the missile 100 and/or the target 104 uses the piece-wise average acceleration along the LOS. The time-to-go τ using this technique by Riggs is calculated according to:
                              τ          =                                    2              ⁢              r                                                      v                                  c                  ⁢                                                                                                    +                                                                    v                    c                    2                                    +                                      4                    ⁢                                          a                      m                                        ⁢                    r                                                                                      ,                            Eq        .                                  ⁢        8            where vc=−{dot over (r)}, the closing velocity, and am is the piece-wise average acceleration along the LOS. When am=0, then Eqs. 7 and 8 are the same. If am is known, then the time-to-go can be obtained directly from Eq. 8. If am is not known, the piece-wise constant acceleration is approximated as:
                                          a            m                    =                                                                      a                  max                                ⁡                                  (                                                            t                      e                                        -                                          t                      0                                                        )                                            +                                                a                  min                                ⁡                                  (                                                            t                      f                                        -                                          t                      e                                                        )                                                      τ                          ,                            Eq        .                                  ⁢        9            where t0 is the initial time, tf is the terminal time, te is the thrust-off time, amax is the average acceleration when the thrust is on from t0 to te, and amin is the average acceleration (actually deceleration) primarily due to drag when the thrust is off from te to tf. Since the time-to-go estimate is a function of am and am is a function of time-to-go, an iterative solution is required.